Prototypical examples for such ill-posed PDEs are provided by the fundamental equations of fluid dynamics, i.e., the incompressible Euler and Navier-Stokes equations as well as the compressible Euler equations. In particular, for the incompressible Navier-Stokes equations, there are currently no global well-posedness results in three space dimensions [1]}. Although admissible weak solutions exist, the uniqueness, stability and regularity of such solutions are outstanding open problems. Similarly for the incompressible Euler equations, well-posedness results are only available in two space dimensions and with regular enough initial data (bounded initial vorticity). Well-posedness results in 2-d with less regular yet physically relevant data such as vortex sheets or in 3-d are mostly unavailable. In fact, it is now known that admissible weak solutions of the incompressible Euler equations need not be unique [2]}. The compressible Euler equations are canonical examples of hyperbolic systems of conservation laws [3]}. Again, there are no rigorous well-posedness results for hyperbolic systems of conservation laws in either two or three space dimensions. Thus, one cannot apply the abstract framework of [4]} (or the recent modifications of [5]}, [6]}) to rigorously conclude that the Bayesian inverse problem for these fundamental PDEs of fluid dynamics is well-posed.
